How to take the double integral of a data set with respect to time

[Leo Liu]

Question: Suppose I have a data file for the acceleration of an object after every $\Delta t_i$, how do I obtain the displacement of it?

Context: Integral in a PID loop, although not exactly what I am asking as one is sum of error: $$\int_0^T \int_0^T \ddot {\vec \theta(t)}dtdt$$
the other is displacement:
$$\int_0^T \left(\dot{\vec \theta(0)}+ \int_0^T \ddot {\vec \theta(t)}dt\right)dt$$

PS I found http://mathforcollege.com/nm/mws/gen/07int/mws_gen_int_spe_trapdiscrete.pdf but it is for single integral over discrete data.

 

[FactChecker]

There are a variety of numerical integration algorithms. If you have a particular scientific subroutine package, you should check whether it has an appropriate subroutine. If you are using Excel, this is a simple example

 

[Leo Liu]

There are a variety of numerical integration algorithms. If you have a particular scientific subroutine package, you should check whether it has an appropriate subroutine. If you are using Excel, this is a simple example

Can I do this on an Arduino? It needs to be computed real time.

 

[FactChecker]

Can I do this on an Arduino? It needs to be computed real time.

That seems very different from integrating data in a file.

 

[Baluncore]

Can I do this on an Arduino? It needs to be computed real time.

Yes.
Keep a running total of the acceleration, which is velocity.
Keep a running total of the velocity, which is displacement.

The problem you will have is that a zero error in the acceleration will accumulate until your displacement exceeds some sensible limit. There will need to be some zero restoration algorithm introduced. You have two unknown constants of integration at the start.

 

[Leo Liu]

The problem you will have is that a zero error in the acceleration will accumulate until your displacement exceeds some sensible limit. There will need to be some zero restoration algorithm introduced.

Could you elaborate, please?

 

[Baluncore]

Errors accumulate with time.
A DC offset error in the acceleration will always bias and drive the velocity one way.
You will need to detect and cancel that error somehow, by knowing the actual displacement at a zero crossing or some ± reference or limit.

 

[Leo Liu]

One more question:
Suppose the acceleration $\ddot x$ is approximately constant during a finite $\Delta t$, is the displacement due to this contribution best described by $\frac 1 2 \ddot x (\Delta t)^2$ or $(\ddot x \Delta t)\Delta t$?